G Allen Gary and Da - NASA...G Allen Gary and Da vid Alexander In tro duction The Suns corona is dominated b y magnetic eld This magnetic eld serv es to conne coronal plasma to predominan

Constructing the Coronal Magnetic Field

By Correlating Parameterized Magnetic Field LinesWith Observed Coronal Plasma Structures

G� Allen Gary �Space Science Laboratory�ES��NASA�Marshall Space Flight Center�AL �� U�S�A�

David Alexander yLockheed Martin Solar and Astrophysics Lab� Org� H�� B�� Hanover St�� Palo Alto� CA �� U�S�A�

�Received �� Accepted in �nal form ��

Abstract� A method is presented for constructing the coronal magnetic �eld fromphotospheric magnetograms and observed coronal loops� A set of magnetic �eldlines generated from magnetogram data is parameterized and then deformed byvarying the parameterized values� The coronal �ux tubes associated with this �eldare adjusted until the correlation between the �eld lines and the observed coronalloops is maximized� A mathematical formulation is described which ensures that �i�the normal component of the photospheric �eld remains unchanged� �ii� the �eld isgiven in the entire corona over an active region� �iii� the �eld remains divergencefree� and �iv� electric currents are introduced into the �eld� It is demonstrated thata parameterization of a potential �eld� comprising a radial stretching of the �eld�can provide a match for a simple bipolar active region� AR �� which crossedthe central meridian on �� Nov �� The result is a non�force�free magnetic �eldwith the Lorentz force being of the order of �� gm cm s�� resulting from anelectric current density of � �� A�m�� Calculations show that the plasma betabecomes larger than unity at a relatively low height of � ��r� supporting the non�force free conclusion� The presence of such strong non�radial currents requires largetransverse pressure gradients to maintain a magnetostatic atmosphere� required bythe relatively persistent nature of the coronal structures observed in AR �� Thisscheme� is an important tool in generating a magnetic �eld solution consistent withthe coronal �ux tube observations and the observed photospheric magnetic �eld�

Key words� Solar� Coronal Loops� Solar� Magnetic Fields� Solar� X�Rays

Abbreviations� MSFC � Marshall Space Flight Center

� E�mail� allen�gary�msfc�nasa�govy E�mail� alexander�lmsal�com

� G� Allen Gary and David Alexander

�� Introduction

The Sun�s corona is dominated by magnetic �eld� This magnetic �eld

serves to con�ne coronal plasma to predominantly linear structures or

loops� the emission of which is pre�eminent in X�ray or EUV images

�e�g� Acton� �� Rosner� Tucker and Vaiana� �� The photospheric

and coronal magnetic �eld also plays a critical role in the heating and

dynamics of the con�ned plasma� resulting in transient phenomena such

as prominence eruptions� coronal mass ejections and �ares� Despite the

overwhelming agreement that the �eld dominates in the low corona� all

evidence for this contention is generally inferred from the form of the

plasma emission� since no direct measure of the coronal �eld can yet

be made �cf� White and Kundu� ��

A standard approach to determining the coronal �eld is the extrap�

olation of the observed �eld at the surface into the corona via various

assumptions about the possible nature of the �eld� e�g� current�free or

force�free �Sakurai� �� Amari et al�� This approach allows

one to generate an approximation to the coronal distribution of mag�

netic �eld which can then be compared with the observed distribution

of the plasma as determined from EUV and X�ray observations �cf�

Jiao� Mclymont and Miki�c� �� While apparently straightforward

and reasonably successful� the extrapolation techniques are subject to

PaperRev��tex� �� no v�� p�

Constructing the Coronal Magnetic Field �

numerous di�culties which are as yet not fully resolved �see Amari et

al��

In particular� the general magnetic �eld associated with an active

region cannot be obtained since there is no closed form solution for

the non�force�free� magnetohydrodynamic problem� This di�culty in

modeling the magnetic �eld is imposed in part by the fact that the

magnetic �eld lines originate in a high�beta plasma region� � � �� in

the granular and inter�granular photosphere and traverse a low�beta

region� � �� in loop structures in the low corona� to culminate once

more in a large�scale � � � region in the solar wind� While there has

been a number of current�free and force�free numerical models which

attempt to extrapolate the magnetic �eld from the photosphere� the

problem is generally ill�posed and assumes � �� Sakurai� ��

Gary� �� McClymont� Jiao� and Miki�c� �� In addition� the upper

boundary conditions required to make the problem tractable and well�

posed are essentially unknown� Consequently� a number of approximate

solutions and numerical methods have been devised each with its own

strengths and weaknesses�

One such assumption is the inclusion of a source surface which mim�

ics the e�ect of the� � � �� solar wind by forcing the �eld to be radial

beyond a certain radius� r � rs� This approach was �rst introduced by

Altschuler and Newkirk �� and Schatten� Wilcox and Ness ��

PaperRev��tex� �� no v�� p��


and has since been used by many authors �e�g� Levine� Schulz and Fra�

zier� �� Sakurai� �� Zhao and Hoeksema� �� As an example� in

potential �eld models� applying the source�surface assumption allows a

spherical potential solution to be derived using the appropriate Green�s

functionG�r� r� and the observed line�of�sight �eld�Bp �Sakurai� ��

In this case the coronal magnetic �eld is given by

B�r� � r��r � r

�ZSG�r� r� Bp�r

� ds��

��

However� it should be noted that in Sakurai�s approach� where G�r� r�

is given by a truncated series� the vector �eld at the source surface is

only approximately radial for a closed�form Green solution�

The general application of a source surface induces a radial stretch�

ing of the �eld lines as compared to a set of potential �eld lines with the

e�ect that the stretching depends on the height of the source surface�

rs� chosen to be spherical in most formulations� The motivation for the

present work comes� in part� from the inability of the source surface

models to provide the necessary modi�cation of the magnetic �eld to

reproduce the observed coronal structures� Speci�cally� a more direct

transformation is required to provide su�cient radial extensions of the

�eld� An alternative approach would be the application of a more gen�

eral� e�g� non�spherical� source surface which has proved successful in



simulating the expected physical consequences of coronal MHD e�ects

in the large�scale coronal �eld �Levine et al�� However� the inclu�

sion of such a source surface has severe implications for the form of the

global �eld and it is not clear if these can be made self�consistent with

the local e�ects required to simulate an active region�

The application of a source�surface or some other means of modify�

ing the �eld in lieu of considering the actual MHD equations� while

mathematically convenient and providing a magnetic �eld solution�

cannot fully address the nature of the problem� The solar wind is a

region of high�beta plasma and the transition from the low�beta corona

to this region is generally more complex than the simple� single radial

shell imposed by the source�surface models� The inclusion of magne�

tohydrodynamic e�ects is clearly an important consideration in the

modeling of the solar corona� particularly for the large�scale structures

seen in white�light� this is beautifully illustrated by the MHD model

of Pneuman and Kopp �� which incorporates the interaction of

�eld and plasma as the coronal material expands out into interplan�

etary space� Consequently� a general solution would require full use

of the MHD equations to address the form of the �eld in the vari�

ous beta regions �see Miki�c� Barnes� and Schnack� �� Jiao et al��

�� Amari et al�� The limiting factor in deriving fully three�

dimensional MHD solutions is the signi�cant amount of computational


G� Allen Gary and David Alexander

e�ort required as well as the need to include all the necessary magnetic

and hydrodynamic boundary conditions �cf� Antiochos and Dahlburg�

��

Most of these studies have concentrated on modeling the corona on

large scales� reproducing the general shape of the streamer belt dur�

ing solar minimum conditions� However� observations from the Yohkoh

spacecraft have shown that the typical �eld con�guration displayed by

the helmet streamer� a closed �eld region surrounded by open �eld with

a cusp�current sheet as the interface between the two� has also been

observed on smaller scales in active region structures �e�g� Tsuneta�

�� suggesting a similarity in the distribution of currents and �eld on

these very di�erent scales� Such a con�guration on the scales of active

regions is not easily addressed by a source surface approach which is

restricted by the form of the global coronal �eld� The objective of the

work presented here is to investigate the interaction between the active

region �eld and the plasma it contains� by reconstructing the coronal

�eld using a novel approach which introduces a radial transformation

of the �eld without resorting to a source surface or computationally

intensive MHD calculations�

We determine the coronal distribution of the �eld by modifying an

initially simple con�guration to produce a distribution which matches

the observed coronal loops� The coronal magnetic �eld is reconstruct�



ed via a transformation procedure in which an initially current�free

distribution of magnetic �eld lines is deformed by performing a radial

stretching at each point of the �eld� Assuming the magnetic �eld to

be traced by the coronal emission� the �eld lines are �stretched� until

the correlation between them and the observed coronal loops is maxi�

mized� In essence� we are perturbing an initially derived potential �eld

subject to the constraints that the three�dimensional coronal �eld is

given by the distribution of X�ray emitting plasma� The form of the

�eld generated in this way requires the presence of cross��eld currents

in the corona to reproduce the observed active region structure� This

indicates that magnetic �eld energy �derived from the line�of�sight �eld

only is not su�cient to produce the active region corona�

The radial stretching transformation used here is directly analogous

to that used in the magnetostatic models of the large�scale corona �see

Bogdan and Low� �� and consequently several implications from

these models can be applied� The use of such a transformation in mag�

netic �eld reconstructions of active regions is� we believe� new and yields

a number of interesting consequences�

The role of the cross��eld currents in heating the corona and their

implications not only for producing the observed active region mor�

phology but also for increasing the free energy available for release in



transient phenomena� provide further impetus to the present investiga�

tion �cf� Wolfson and Dlamini� ��

We introduce our transformation method in section � and apply it

to the case of AR �� in section �� The results are interpreted in

section � where we discuss the implications of the inferred non�radial

currents� The correlation method between the theoretical and observed

coronal �eld is given in Appendix A� while the magnitude of the inferred

currents is calculated in Appendix B� Our conclusions are presented in

section ��

�� Transformation of the Magnetic Field

The analysis proposed here is based on a transformation method which

assumes � �� the conservation of magnetic �ux r � B � �� an

invariant photospheric vertical magnetic �eld� i�e�� the observed radial

boundary condition Br�r � r� is unchanged� �� the initial con�gura�

tion is a known �potential �eld� r � B � � and r� B � � � i�e�� the

unperturbed� unstretched� zero�order �eld� �� a transformation is per�

formed which radially stretches the �eld lines �Fi� i � all �eld lines �

and simulates possible MHD e�ects in a simple analytic variation of

only one of the spherical coordinates� Fi represents the locus of the

points which de�ne the ith �eld line�



The point�by�point spatial transformation TK of the �eld lines is

given by

TKfFi�r� �� g � Fi�k�r � r� � r��

which is a function of the height� h � r � r�� and the constant radial

stretching factor k� Using this transformation from the potential �eld

�B and r � B � r � B� � �� the resulting magnetic �eld �B� can be

shown to be

B�r � w�r� r�� k Br� ��

B�� v�r� r�� k B��

B�� v�r� r�� k B��

where

w�r ��h� r��

�kh� r��and v�r �

h� r�k �k h� r��

� �

The parameter k � � implies no stretching� and k � � implies an

increasing �eld stretching as r increases� However� we emphasize that

at r � r� �h � � the radial component of the photospheric �eld is

recovered unchanged for all values of k� A similar �stretching� transfor�

mation was introduced in modeling the large�scale corona above helmet


�� G� Allen Gary and David Alexander

streamers �e�g� Bogdan and Low� �� Gibson and Bagenal� �� Gib�

son and Low �� have recently utilized a generalized radial solution

for stretching the magnetic �eld in the modeling of CMEs� This general�

ized solution invokes a coordinate transformation such that r �� kr�a

where k and a are arbitrary constants �Gibson and Low choose k � �

without loss of generality � Such a transformation was required for the

magnetic �eld to satisfy the magnetostatic equations� This transforma�

tion is equivalent to that shown in equation �� with a � ��k�� r� and

k � �� although in the form adopted by Gibson and Low it becomes a

translational coordinate transform rather than the rescaling transform

adopted here�

The electric currents induced by the stretching of the magnetic �eld

are easily determined from

J�r� �� r�B�

��

r sin�

��

��B�

� sin� ��

��B�

�

��ur

��

r

��

sin�

�

��B�

r ��

�r�rB�

�

��u�

��

r

��

�r�rB�

� ��

��B�

r

��u��

which are orthogonal to the radial direction� since� the coe�cient of �ur�

��

��v�r B� sin� �

�

��v�r B�

��


Constructing the Coronal Magnetic Field ��

These cross��eld currents represent the physical mechanism which pro�

duces the three�dimensional distribution of the magnetic �eld required

to match the observed active region morphology� The inferred presence

of these non�radial currents have strong implications for the force bal�

ance in the active region corona� The magnitude of these currents and

their corresponding Lorentz forces will be considered in section ��

�� Data Analysis� Active Region ��

The physical cynosure for this study� is the active region AR ��

This region has the following desirable properties� �� the photospher�

ic magnetic �eld is a relatively simple bipolar con�guration� �� the

region is at a slow evolutionary phase when observed at central merid�

ian passage� �� it is the only active region in the observed hemisphere�

�� there is a large data set including the observed coronal loops from

Yohkoh�SXT and SOHO�EIT instruments�

An SXT image of AR �� as it crossed the Central Meridian on

�� Nov � �� UT is shown in Figure �� One can see that the

active region is compact and exhibits a relatively simple bipolar struc�

ture� Each frame of Figure � shows the same soft X�ray image� A com�

parison between the extrapolated �eld lines and the soft X�ray loops

clearly demonstrates that a potential �eld �k � � does not adequately



reproduce the observed structure� The �eld lines were generated from

the spherical potential code of Sakurai �� using photospher�

ic �eld data from a KPNO magnetogram taken at �� UT� The

�eld lines have been rotated to coincide with the time of the soft X�ray

image and a heliographic grid �� degrees has been superposed to

show the location and size of the image� As the stretching parameter�

k� is increased� the match to the observed X�ray structure improves� It

is apparent from Figure � that a stretching factor of k � � best approx�

imates the distribution of coronal loops� In Appendix A� we derive the

best �t to be k � �� Even though we do not have a stereoscopic view

to establish that the �eld lines are consistent in height with the coronal

loops� we can use the solar rotation with the same �eld lines to estab�

lish the credibility of the correlation� The results of �eld lines for the

succeeding days are shown in Figure ��

Therefore� by these correlations� it is seen that the radial stretching

of the �eld lines provide a signi�cant improvement in matching the

soft X�ray �ux tubes for this active region� We have provided a visual

comparison of the e�ect and numerical veri�cation of the improvement�

In the next section we will interpret this important �nding�



�� Interpretation of the Radially Stretched Field Lines

Field�Plasma Connection

We can estimate whether a linear stretching factor of �� is rea�

sonable by comparing with the results of MHD models� Pneuman and

Kopp �� developed a numerical MHD model of the large�scale coro�

na to describe the interaction between the magnetic �eld and coronal

plasma in the development of a helmet streamer� The application of

this model results in a radial stretching of the �eld as compared to a

potential �eld and the formation of a current sheet� Using Figure � of

Pneuman and Kopp �� we can estimate the spatial stretch imposed

by their numerical MHD model� In the height range� r � �� r��

the corresponding radial stretching factors would be k � ��

similar to the results obtained from Figure �� The MHD results of

Pneuman and Kopp �� do not provide for a large degree of radi�

al stretching close to the solar surface �h � �� r� as required by

the active region studied here� However� they were not concerned with

the detailed modeling of the corona on active region scales� adopting

an essentially potential �eld distribution low in the corona� In order

to produce the required radial stretching in the active regions while

maintaining the large�scale coronal geometry� a much more complex

interaction between the plasma and the magnetic �eld is necessary�



Cusped loops are widespread� seem to have several di�erent man�

ifestations� and have been observed at heights of �� r� which give

further evidence for the in�uence of MHD e�ects at heights consid�

erably lower than previously considered �Hiei and Hundhausen ��

Strong �� These observations all imply a departure from the clas�

sical potential �eld model and� consequently� stress the importance of

currents in the corona�

From the Pneuman and Kopp �� results� at r � �� r�� the

plasma beta can be estimated to be � � �� and hence far from

any force�free approximation �Suess� �� indicating the probability

that cross��eld currents exist in the large�scale coronal �eld� Lower in

the corona one expects the plasma beta to be very much less than unity�

This is not always true for active regions� however� In the outer bright

loops of a large active region� an estimate of the plasma beta can be

determined via

� �plasma pressure

magnetic pressure�

��nkT

B��

n�T�B��

� ��

where B�� B�� G� n� n�� cm�� and T� T��K� For

reasonable active region parameters� �B�� n� � �� T� � ��

Thus� for a typical active region� we expect that the gas pressure is

not completely ignorable� In Figure � we show the magnitude of the



magnetic �eld strength along the potential �eld lines �k � � as a

function of height� This shows that at few tenths of r� the magnetic

�eld does indeed fall below ��G�

Secondly� using the form of the magnetic �eld with height shown in

Figure � in the case of AR �� we can estimate the plasma beta at

each point in the active region� To do this we consider the active region

when it was at the West limb �� Dec � and determine the emission

measure and temperature from the Yohkoh�SXT using a �lter�ratio

technique �see Hara et al�� This was done by �rst summing ��

images in each of the two thin �lters of the SXT to provide a single

summed image pair �the images were co�registered and corrected for

background before summing see Alexander� �� For each image in

the image pair� active region pixels lying along the arc de�ned by a

given height above the limb were summed and the ratio of the two

�lters taken to provide an average temperature and emission measure

for the active region at that height� The density is then determined

from the emission measure assuming a line�of�sight depth of ��

km� i�e� n �pEMAl where A is the area of an SXT pixel ��

km� � Armed with the average density� temperature and magnetic �eld

variation with height we determined the plasma beta variation shown

in Figure �� The plasma beta reaches unity at a height �� km

or �� r��



Electric Currents in the Corona

The modi�cation of the coronal magnetic �eld distribution necessary

to support this active region con�guration requires the presence of a

coronal electric current system� An approximation to the magnitude of

the induced electric currents can by made from equation �� by assum�

ing that the spherical coordinate system locally can be replaced by

a Cartesian coordinate system �see Appendix B � This approximation

yields the ��component of the electric current as J� � �� A m��

at z � ��r�� The magnitude of this current is comparable to that

found in magnetostatic models of the large�scale corona �e�g� Bage�

nal and Gibson� �� where typical coronal current densities of order

�� A m�� are found at heights� h � �� r�� The geometry

assumed in the large�scale coronal models is relatively simple� especial�

ly at low heights with assumed �eld strengths �� G and so we should

not expect the conditions to be too similar to the active regions being

modeled here� However� these models and our results indicate that large

coronal current densities are a common feature of the corona�

A simple calculation will show that these electric currents are some�

what high compared to the presure force required for coronal force

balance� where the electric currents are on the order of �� A m��

This can be seen by deriving the equivalent electric current correspond�

ing to the radial pressure gradient� Assume the radial component of



the pressure gradients is balanced by the gravity force at r � ��r�

�z � ��r� �

jdp

drj � �g � �mpn gsurf

�r

r�

�� n� erg cm��

where gsurf � �� cm s�� is the gravitational acceleration at

the surface of the Sun� Then let the Lorentz force per unit mass for a

current density of JL be equal to this gravity force�

jJL �Bj � JLB � �� erg cm��

and for B � �� G� JL � �� G cm�� which implies that

JL � �� Am�� cf�� Priest� �� p� �� eqn �� Here we have

used �� Gm�� A m�� and erg cm�� G�� This value

for JL is signi�cantly smaller than J� estimated above�

The electric currents induced by radial stretching are larger than

would be required to overcome the gravity term in the force equation�

However� these currents are consistent with those of Bogdan and Low

�� who showed that a �D magnetostatic model of the solar coro�

na with a �r� gravity �eld can be generated analytically with the

assumption that the coronal currents be everywhere perpendicular to

gravity �i�e� non�radial � This is the identical situation implied by the

�eld transformations of section �� Consequently� we envisage that the



large Lorentz forces generated by the coronal currents would be bal�

anced primarily by transverse pressure gradients� i�e� dpd�� Such a

model is beyond the scope of the present work� since a proper treat�

ment of an active region atmosphere would require the self�consistent

incorporation of an energy equation in addition to the force balance

equations�

In the foregoing� we have neglected the possibility of MHD �ows

providing the necessary force balance� However� a quick calculation

suggest that in the absence of substantially enhanced �ows the force

balance must be dominated by the transverse pressure gradients as

stated above� For steady out�ows� the equation of motion has the addi�

tional term nm�v � r v� As an estimate of the order of magnitude of

this term we use the results of Grall et al� �� for polar winds� For

a ��km s�� velocity increase over � r�� the solar wind e�ect is on the

order of ��erg cm�� for a mass density of � � ��g cm��

comparable to jdpdrj � �g

The electric currents required to generate the active region morphol�

ogy place severe constraints on the forces present in the corona� How�

ever� without more understanding of the gradients across the observed

loop structures in the active region and the magnitude of the �ows

within these structures we cannot rule out such large cross��eld cur�

rents�



We should note that we chose the lower boundary condition for

the potential solution to be the observed magnetogram with the tacit

assumption of an upper boundary condition of zero �eld at in�nity� In

fact� what we have not considered is the possibility that the presence of

MHD �ows deforms the actual upper boundary condition� a situation

which the application of the source�surface models addresses� Hence�

in future work we will include a potential� source�surface model to

augment the present approach�

�� Conclusion

The results of this study are� �� radially stretching extrapolated poten�

tial �eld lines improves the relationship between the coronal �eld

and the observed soft X�ray structures� �� The resulting magnetic

�eld is non�force free with non�radial electric currents� Though not

shown explicitly� we have determined that a truncated Green�s func�

tion source�surface model will not provide adequate radial stretching for

the region for any source�surface height �Sakurai� �� Magneto�

static force balance can only be achieved by strong transverse pressure

gradients�

We conjecture that the cross��eld currents simulate the MHD solar�

wind e�ect where the plasma beta � � �pB� � �� by altering the



�eld to simulate a solution with the appropriate upper boundary condi�

tions� In addition� the presence of such non�force�free distributions may

yield the necessary �latent� coronal energy for coronal mass ejections�

and may be responsible for the energy released in solar �ares�

The resulting magnetic �eld solution provides an important con�

struct to which other analytical and numerical models must comply�

We have derived a coronal magnetic model by establishing a magnetic

�eld which yields a set of �eld lines consistent with the observed coronal

loops� However� the initial �eld was restricted by the use of a potential

�eld extrapolation� The code used to generate the extrapolated �eld

is further restricted in that the locations for conjugate footpoints of

a closed coronal �eld line are �xed� Future extension of the process

would incorporate a more general force�free model in which the pair

of conjugate foot points can be varied� providing an environment in

which a general �ux tube could be realized� Increasing the number of

variables would allow a better �D magnetic �eld solution without loss

of information�

Acknowledgements

Part of this work was supported by the NASA O�ce of Space Science

under the space science theme of the Sun�Earth Connection �GAG and



under contract NAS �� DA � We thank S� Suess and R� Moore

�MSFC and S� Gibson �GSFC for helpful comments and discussions�

We wish to thank T� Sakurai for making his potential code available

and his assistant in its implementation at MSFC� Yohkoh is a mission

of the Japanese Institute for Space and Astronautical Science�

Appendix

A� Coronal Flux Correlation with Stretched Field Lines

In this Appendix� we establish a quantitative correlation between the

observed coronal loops and the extrapolated �eld lines� The following

steps are employed� �� The coronal loops of a combined set of SXT

images for �� November � are enhanced by an unsharp masking

algorithm� �� Individual loops are delineated and their loci determined

�� The position of the average apex of the coronal loops are compared

to the position of the average apex of the stretched �eld lines for a series

of stretching factors� The magnetic �eld apex is de�ned as the position

of the maximum height of stretched magnetic �eld lines� The line�of�

sight loop apex is de�ned as the position which has the maximum

radial value� rmax� �in �D from the centroid of the distribution of

longitudinal magnetic �eld strength� The result is that the stretching



factor of k � �� provides the maximum correlation for coronal loops

�in the range �� rmax � �� arcsec or ��r� � rmax � ��r�

A set of nine SXT images over a three hour interval were summed

after co�registration using a transformation for solar di�erential rota�

tion �Scherrer� Wilcox� and Svalgaard� �� The Yohkoh�SXT AlMg

images� shown in Figure �� were taken over a � hour period from ��

to �� UT with the �rst image being the reference image to which

all others are co�aligned� The enhancement process of unsharp mask�

ing� subtraction of a smoothed version of the image from itself� was

applied to the summed images and the result is displayed in Figure �b�

The unsharp masking algorithm uses a Lee Filter which smoothes addi�

tive image noise by generating statistics in a local neighborhood and

comparing them to expected values �Lee �� IDL routine LEEFILT �

From the enhanced image �� coronal loops were easily identi�ed Figure

�c�

To locate the apexes in the image plane� a polar coordinate system

was de�ned by locating the origin at the centroid of the longitudinal

�eld strength distribution� The polar angle for a line from the origin to

an apex is measure from a line parallel to the vector from disk center

to the extreme western limb� The magnetic centroid was transformed

to the appropriate image coordinates taking into account the time dif�

ference between the magnetogram and SXT reference time using the



sidereal di�erential rotation rate� The magnetic �eld lines are extrapo�

lated from the KPNO magnetogram and� hence� the magnetic centroid

provided a natural origin for the polar coordinates of the �eld lines for

this magnetically simply�con�gured active region�

The radial apex position for short loops �r � �� arcsec can be

signi�cantly displaced from an apex as de�ned by the maximum height

for the computed �eld lines� The angular position of the apex can be

shown to be a limiting envelope of the radial apex angles for the com�

puted �eld lines� Hence� the apex of the �eld lines is compared with the

radial apex of the coronal loops� since the coronal loops show no large

distortion that would result from line�of�sight e�ects this implies that

the coronal loops have a low inclination angle with respect to the tan�

gential plane� The maximum coronal loop apex has a radial value of ��

arcsec from the magnetic centroid� Therefore� we have limited the apex

comparison to the radial range �� arcsec� The coronal loops were

classi�ed into northern and southern groups since their polar angles are

separated by � �� The average apex for these two groups is shown

in Figure � The sign of angular position for the southern groups and

�eld lines have been reversed for this display�

The average height apex in the comparison range is calculated for

a set of stretching factors �k�� A total of ��

unstretched� potential� �eld lines �k�� were calculated using the



Sakurai spherical potential code and the stretching transformation was

applied to these �eld lines �Sakurai� �� This distribution of the

�eldline footpoints was determine by the code�s magnetic �ux strength

algorithm� The resulting angular coordinate for the average radial apex

as a function of the stretching factors is shown in Figure � The average

apex of the southern set of stretched �eld lines intercept the value of

the average coronal apex at k � �� and the northern set at �� with a

mean of k � �� Hence� �eld lines with a stretching factor of k � ��

best approximate the structure of the active region corona�

B� Approximation of J�

In this Appendix� an approximation to the ��component of the electric

current density J� is derived by approximating the equivalent Jx� The

spherical coordinates are replaced locally with the appropriate Carte�

sian expressions� For Jx� we have

��Jx ��B�

y

�z��B�

z

�y� ��

and for the potential case� k � �� Jx � �� The spatial stretch gives a

Jx value of



��Jx ��v�z� r�� k By�

�z��kv�z� r�� k � Bz�

�y� ��

where we have replaced w by its equivalent� w � k�v�� Hence

��Jx � �v�k�v� �By

�z�By

�v

�z��k�Bz

�v�

�y�k�v�

��By

�z��Bz

�y

��

and the last two terms are zero� For k � �� and z � ��r� then

�v�z � r�� k �k�kz � r� � � ��r� and �v � k�v� � ��

hence j��Jxj � ��By�z� Now �By�z � �Bz�y � Bz�z �

��r� L and L � �� m is the scale of the active region as

measured from the Yohkoh�SXT images� Assuming a �eld strength of

�� G at z � ��r�� j��Jxj � �� Gm��

�� Am��

Hence� the ��component of the electric current density J� �

��A m��

List of Figures

Figure �� Transformed �eld lines in relationship to coronal loop

structures as the radial stretching factor is varied from k � ��

�� to �� for ��Nov�� An arrow is used to draw attention to the

evolution of the transformed �eld lines in the south of the active region

which conform to the observed loops as k is increased�



Figure �� Transformed �eld lines in relationship with coronal loop

structures for the radial stretching factor k � �� for ��Nov ��

�� and Dec�� limb view �

Figure �� A plot of the magnitude of the magnetic �eld strength �G

along the potential �eld lines versus height� For reference the dashed

curves are a plot of the empirical functions B � �� e��ph�where

� � �� and ��

Figure �� The inferred variation of gas pressure and plasma beta

with height for AR �� A line�of�sight integration path length of

�� km was assumed� The median magnetic �eld was given by the

expression of Figure �� B � �� e��ph�

Figure �� Enhancement of the Yohkoh�SXT Image for AR �� The

three panels show �a the original summed image of nine Yohkoh�SXT�

�b the enhance coronal loop image using an unsharp masking tech�

nique� and �c the digitized coronal loops using a cursor input and

then employing a polynomial �t on the polar coordinates of the points

to smooth the curves�

Figure � The angular polar coordinates of the average apex posi�

tion for the average northern �dashed and southern �solid �eld lines

as function of the stretching factor� These curves are compared to the

average radial apex position for the coronal loops �horizontal lines �

The southern �northern average of the coronal loops of �� cor�



responds to stretching of k � �� The mean value of k � �� is

the average stretching factor for the entire region�

References

Acton� L� W� ��Comparison of Yohkoh X�ray and Other Solar Activity Param�eters for November �� to November �� Proc� of Ninth Coolstar Workshop�Florence� Italy� �� October ��

Alexander� D�� Temperature Structure of the Quiet X�ray Corona�� J� Geophys�Res�� in print�

Altschuler� M� D�� and Newkirk� G� Jr�� Magnetic Fields and the Structure ofthe Solar Corona I� Methods of Calculating the Coronal Fields�� Solar Physics�Vol� �� pp� ��

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Antiochos� S� K� and Dahlburg� R� B�� The Implications of �D for Solar MHDModeling�� Solar Physics� Vol� �� pp� ��

Bagenal� F�� and Gibson� S� E�� Modeling the large�scale structure of the solarcorona�� Journal Geophys� Research� Vol� �� pp� ��

Bogdan� T� J�� and Low� B� C�� The three�dimensional structure of magneticatmospheres II� Modeling the large�scale corona�� Astrophys� Journal� Vol� ��pp� ��

Grall� R� R�� Coles� W� A�� Klinglesmith� M� T�� Breen� A� R�� Williams� P� J� S��Markkanen� J� and Esser� R�� Rapid Acceleration of the Polar Solar Wind��Nature�Vol� �� pp� ��

Gary� G� A�� Concerning the Extrapolation of Solar Non�Linear Force�FreeMagnetic Fields�� Memorie della Societa Astronomica Italiana�Vol� �� pp��

Gibson� S� E�� and Bagenal� F�� The large�scale magnetic �eld and densitydistribution in the solar minimum corona�� Journal Geophys� Research� Vol�� pp� ��

Gibson� S� E�� and Low� B� C�� A Time�Dependent Three�Dimensional Magne�tohydrodynamic Model of the Coronal Mass Ejection�� Astrophys� Journal� Vol�� pp� � ��

Hara� H�� Tsuneta� S�� Lemen� J� R�� Acton� L� W�� and McTiernan� J� M�� High�temperature plasmas in active regions observed with the Soft X�ray Tele�scope aboard Yohkoh�� Publ� Astr� Soc� Japan� Vol� �� pp� L��L�

Hiei� E�� Hundhausen� A� J�� Development of a Coronal Helmet Streamer of�� January �� in Magnetodynamic Phenomena in the Solar Atmosphere �Prototypes of Stellar Magnetic Activity� eds� Y� Uchida� T� Kosugi� and H� S�Hudson� Kluwer Academic Pub�� Boston� pp� ��

Jiao� L�� McClymont� A� N�� and Miki�c� Z�� Reconstruction of the Three�Dimensional Coronal Magnetic Field�� Solar Physics� Vol� �� pp� ��

Lee� X� X�� Speckle Suppression and Analysis for Synthetic Aperture Radar��Optical Engineering�Vol� �� pp� ��

Levine� R� H�� Schulz� M�� and Frazier� E� N�� Simulation of the MagneticStructure of the Inner Heliosphere by Means of a Non�Spherical Source Surface��Solar Physics�Vol� �� pp� ��



McClymont� A� N�� Jiao� L�� and Miki�c� Z�� Problems and Progress in Comput�ing Three�Dimensional Coronal Active Region Magnetic Fields from BoundaryData�� Solar Physics� Vol� �� pp� ��

Miki�c� Z�� Barnes� D�C�� and Schnack� D�D�� Dynamical Evolution of a SolarCoronal Magnetic Field Arcade�� Astrophys� Journal� Vol� � � pp� ��

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Solar Corona�� Solar Physics�Vol� � � pp� �� Rosner� R�� Tucker� W� H�� and Vaiana� G� S�� Dynamics of the Quiescent

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Regions�� Space Science Reviews�Vol� �� pp� ��Sakurai� T�� Computational Modeling of Solar Magnetic Fields� Astronomical

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Address for correspondence� G� Allen GarySpace Science Laboratory�ES��NASA�Marshall Space Flight Center�AL �� U�S�A�



Figure �� Transformed �eld lines in relationship to coronal loop structures as theradial stretching factor is varied from k � �� to �� for ��Nov�� Anarrow is used to draw attention to the evolution of the transformed �eld lines in thesouth of the active region which conform to the observed loops as k is increased�



Figure �� Transformed �eld lines in relationship with coronal loop structures forthe radial stretching factor k � �� for ��Nov �� and Dec�� limb view��



Figure �� A plot of the magnitude of the magnetic �eld strength �G� along thepotential �eld lines versus height� For reference the dashed curves are a plot of the

empirical functions B � e��ph�where � � �� and ��



0 5 10 15 20Height (104 km)

0.0

0.2

0.4

0.6

0.8

1.0

Pla

sma

Bet

a

Figure � The inferred variation of gas pressure and plasma beta with height forAR �� A line�of�sight integration path length of �� km was assumed� The

median magnetic �eld was given by the expression of Figure �� B � e��ph�



Figure �� Enhancement of the Yohkoh�SXT Image for AR �� The three panelsshow �a� the original summed image of nine Yohkoh�SXT� �b� the enhance coronalloop image using an unsharp masking technique� and �c� the digitized coronal loopsusing a cursor input and then employing a polynomial �t on the polar coordinatesof the points to smooth the curves�



Figure �� The angular polar coordinates of the average apex position for the averagenorthern �dashed� and southern �solid� �eld lines as function of the stretching factor�These curves are compared to the average radial apex position for the coronal loops�horizontal lines�� The southern �northern� average of the coronal loops of ��corresponds to stretching of k � �� The mean value of k � �� is the averagestretching factor for the entire region�


G Allen Gary and Da - NASA...G Allen Gary and Da vid Alexander In tro duction The Suns corona is dominated b y magnetic eld This magnetic eld serv es to conne coronal plasma to predominan

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